Concave macroscopic fundamental entropy

For each admitted material constitution $c$, the model adopts a finite scalar function

$S_c:D_c⟶\mathbb{R}\text{ with values measured in }\mathrm{J}{\mathrm{K}}^{-1},$

on the equilibrium state domain $D_c$ of Equilibrium state domains. This framework treats $S_c$ as primitive constitutive data: its existence and form are assumed, not derived from the word entropy, statistical mechanics, or experimental evidence.

On every specified convex region $C\subseteq D_c$ in the dimensionless coordinates of Equilibrium state domains, $S_c$ is continuous and concave, meaning that for all $x,y\in C$ and every $\alpha \in [0,1]$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length),

$S_c((1-\alpha )x+\alpha y)\ge (1-\alpha )S_c(x)+\alpha S_c(y)\mathrm{.}$

The domain assumptions are local to the material and region named; no unlisted coordinate tuple is presumed physically realizable. The input of $S_c$ is the represented state ${\pi }_c(e)$; when the function is written using physical coordinates $(U,V,n_1,\dots ,n_r)$, those coordinates mean the SI values whose dimensionless numerical tuple is ${\pi }_c(e)$. For independent separated macroscopic systems $A,B$ admitted by Additive separated systems and tagged bulk scaling, entropy is additive under that postulate's composition map ${\kappa }_{AB}$:

$S_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=S_A(x_A)+S_B(x_B)\mathrm{.}$

On each separately designated open, single-phase chart in the physical coordinates $(U,V,n_1,\dots ,n_r)$, assume the numerical representative of $S_c$ is $C^2$ in the sense of $C^k$ Euclidean maps and diffeomorphisms: its first and second coordinate partial derivatives exist and are continuous. The partial derivative $S_U$ at fixed $V,n_1,\dots ,n_r$ is the ordinary partial derivative in the $U$ coordinate, as in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and is assumed finite and strictly positive. The numerical chart input uses $\widehat{U}=U\mathrm{/}(1\mathrm{J})$; consequently $S_U=(1\mathrm{J}{)}^{-1}\mathrm{\partial }{S}_c\mathrm{/}\mathrm{\partial }\widehat{U}$ and has units ${\mathrm{K}}^{-1}$. The other physical coordinates are similarly represented using the fixed SI units of SI quantities and unit conventions. No differentiability, strict concavity, or positive $S_U$ is asserted outside those charts. This item does not itself define a thermometer calibration or claim a temperature law.

For the tagged bulk family with scaling maps ${\delta }_{\lambda }$ and domains $B_{c,\lambda }$ from Additive separated systems and tagged bulk scaling, also adopt, for every admitted $\lambda >0$ and $x\in B_{c,\lambda }$,

$S_{c^{(\lambda )}}({\delta }_{\lambda }x)=\lambda S_c(x)\mathrm{.}$

This is a bulk homogeneity assumption for tagged scaled systems, not exact finite-size scaling for every body. Lieb–Yngvason develop a different accessibility-based route in which entropy is represented from additional operational hypotheses; this framework does not claim that their derivation proves the present primitive entropy postulate.